Basically reiterating things already said...
touqra said:
Can the energy density of an empty but expanding spacetime decrease with expansion?
As some respondents already said, in the context of relativistic cosmology (right?) "empty spacetime" usually means a vacuum solution of the Einstein field equation (EFE) of the general theory of relativity (gtr). By definition, in a vacuum solution, the stress-energy tensor vanishes identically. The energy-density is just one component (evaluated in an appropriate frame field) of this tensor, so it vanishes too.
touqra said:
Does a black hole have a stress-energy tensor?
The simplest and most often discussed models of black holes in gtr are exact vacuum solutions such as the Schwarzschild or Kerr vacuums, so these have stress-energy tensors which are well-defined but which
vanish. (The equation x=0 has a real root, zero, whereas the equation x^2+1 has no real roots--- this analogy suggests why I would pedantically object to any suggestion that a vacuum solution might not possesses a stress-energy tensor at all!)
However, there are other models of black holes (compact objects which possesses event horizons) such as the ingoing Vaidya null dust (1951)
<br />
ds^2 = -\left( 1-\frac{2m(t)}{r} \right) \, dt^2 <br />
+ \, 2 \, dt \, dr<br />
+ \, r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),<br />
<br />
-\infty < t < \infty, \; 0 < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi<br />
where m(t) is an arbitrary
non-negative non-decreasing smooth function, which models a black hole with infalling incoherent radiation, and the Reissner-Nordstrom electrovacuum, which models a static charged black hole). The stress-energy tensors of these solutions do not everywhere vanish.
For those of you learning about black hole formation, especially if you are worried about the odd global structure of the "eternal black hole", the Vaidya null dust is a terrific example. Consider the case of a smooth function m(t) which is initially zero and at some time begins to increase, then levels off at some positive value. This models a region of Minkowski spacetime, until along comes a contracting shell of incoherent radiation (see the discussion in Feynman's Lectures on Physics of a similar thought experiment), which collapses on a point forming a black hole there. Follow the evolution of the event horizon and consider the implications! (I can explain if this doesn't become clear after some thought.)
touqra said:
What about a non-zero cosmological constant?
Another good example: the Schwarzschild-de Sitter lambdavacuum (1918)
<br />
ds^2 = -\left(1- \frac{2m}{r} + \frac{r^2}{a^2} \right) \, dt^2 + \, \frac{dr^2}{1- \frac{2m}{r} + \frac{r^2}{a^2} }<br />
+ \, r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),<br />
<br />
-\infty < t < \infty, \; r_0 < r < \infty, \; <br />
0 < \theta < \pi, \; -\pi < \phi < \pi<br />
where r_0 is the positive real root of r^3 - a^2 \, r + 2 m \, a = 0, models a static spherically symmetric black hole with a Lambda term (as I wrote the solution above, \Lambda=\frac{3}{a^2}, where a is the "radius" of the cosmological horizon). This term contributes a nonzero term to the stress-energy tensor.
For those of you learning about global structure (Carter-Penrose diagrams and all that), here's an exercise: find the Carter-Penrose diagram for the Schwarzschild-de Sitter lambdavacuum. (I can provide a citation to an arXiv eprint which gives the solution.)
touqra said:
Without a non-zero cosmological constant, an empty spacetime wouldn't expand, isn't it?
The Kasner vacuum (1921)
<br />
ds^2 = -dt^2 + \, t^{2a} \, dx^2 + \, t^{2b} \, dy^2 + \, t^{2c} \, dz^2, <br />
\; \;<br />
0 < t < \infty, \; -\infty < x, \, y, \, z < \infty<br />
where
a+b+c = a^2+b^2+c^2 = 1
is a simple homogeneous anisotropic vacuum solution which expands from a Big Bang type singularity. You can take
<br />
a = \frac{-u}{1+u+u^2}, \; <br />
b = \frac{1+u}{1+u+u^2}, \;<br />
c = \frac{u \, (1+u)}{1+u+u^2}<br />
so for example a=-2/7, \; b=3/7, \; c=6/7 is a rational solution of the constraint equations. This exact vacuum solution arises as a limiting case of the Kasner dust solution. (In gtr, a "dust" is a perfect fluid with vanishing pressure; the so-called "matter-dominated FRW models" in cosmology, or better, FRW dusts, are exact dust solutions.)
The Kasner vacuum is a terrific computational example for those of you learning about geodesic equations, or about computing curvature the Cartan way, or about the kinematic decomposition, BTW. It also provides an excellent example of a simple exact solution with extremely interesting "light cones in the large" (see the monograph by Hawking and Ellis,
The Large Scale Structure of Space-Time).
touqra said:
How is that possible if the mass of the BH contributes to a non-zero total energy
Not sure I understand the question, but as pervect (and Stingray) said, in gtr, localizing the energy of the gravitational field itself, is quite tricky. As he said, for certain kinds of spacetimes which model "isolated objects", there are sensible methods of assigning a physically reasonable mass and angular momentum. These include black holes solutions like the Kerr vacuum.
touqra said:
and the volume of the BH can be taken as a spherical volume with a total area equals to that of its event horizon?
No, that wouldn't be correct even for the simplest kind of stellar model, in which a static spherically symmetric perfect fluid ball (these are all known, and simple examples can even be found readily; see the arxiv for recent reviews coauthored by Matt Visser) is matched across the surface (where the pressure vanishes but the density typically does not) to an exterior Schwarzschild vacuum region. The geometric significance of the Schwarzschild radius is that A = 4 \pi \, r_0^2 for the surface area of the sphere t=t_0, \, r=r_0 remains valid, but distance along a radius t=t_0, \, \theta=\theta_0, \, \phi=\phi_0 and volume are given by more elaborate formulas, in the case of such a stellar model.
In the case of a black hole, the best short answer is that the interior of an event horizon is nothing like the interior of a hollow beach ball; roughly, the spacetime is so curved that there is nothing inside which can be given a volume. Maybe the best short explanation of the reason why not would be to say that inside the horizon, no matter can remain static, so unlike the case of the static fluid ball, there is no sensible way of assigning a volume to some static ball-like region of "space" inside the horizon.
Now read again what I said above about the Vaidya solution modeling the black hole formed by collapse of a shell of radiation.
Stingray said:
I'm glad you're here, Stingray!
